# Question 8:

## Part 8(a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Row minima are $-3, -1, -2, -4$; Column maxima are $4, 2, 5$ | M1 | Clear attempt to find row maximin and column minimax; either row minimums or column maximums correct, or at least five of seven values stated correctly |
| As Row maximin $(-1)$ is not equal to Column minimax $(2)$, i.e. $-1 \neq 2$, the game is not stable | A1 | CAO dependent on all rowmins and colmaxs correct; states $-1 \neq 2$; $-1$ clearly identified as row maximin and $2$ as column minimax |

## Part 8(b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Option Q dominates option T | B1 | Correct statement — must include word 'dominate' or exact equivalent |
| Because e.g. $-3 > -4$, $2 > 0$ and $5 > 2$ | B1 | Correct inequalities — must be clear that all inequalities must hold |

## Part 8(c)(i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Augment by $+3$: $\begin{pmatrix}-3&2&5\\2&-1&0\\4&-2&-1\end{pmatrix} \rightarrow \begin{pmatrix}0&5&8\\5&2&3\\7&1&2\end{pmatrix}$ | B1 | Correct augmentation — possibly implied by later working |
| Maximise $P - V = 0$ where $V$ is the value of the augmented game to A | B1 | Correct objective defined |
| $V - 5p_2 - 7p_3 + r = 0$ | M1 | At least three equations in $V, p_1, p_2, p_3$ and at least one dummy variable seen |
| $V - 5p_1 - 2p_2 - p_3 + s = 0$ | | |
| $V - 8p_1 - 3p_2 - 2p_3 + t = 0$ | A1 | CAO |
| $p_1 + p_2 + p_3 + u = 1$ | B1 | Correct probability equation |

## Part 8(c)(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Initial Simplex tableau with correct rows: | M1 | Any two numerically correct rows |
| b.v. \| $V$ \| $p_1$ \| $p_2$ \| $p_3$ \| $r$ \| $s$ \| $t$ \| $u$ \| Value | | |
| $r$ \| $1$ \| $0$ \| $-5$ \| $-7$ \| $1$ \| $0$ \| $0$ \| $0$ \| $0$ | | |
| $s$ \| $1$ \| $-5$ \| $-2$ \| $-1$ \| $0$ \| $1$ \| $0$ \| $0$ \| $0$ | | |
| $t$ \| $1$ \| $-8$ \| $-3$ \| $-2$ \| $0$ \| $0$ \| $1$ \| $0$ \| $0$ | | |
| $u$ \| $0$ \| $1$ \| $1$ \| $1$ \| $0$ \| $0$ \| $0$ \| $1$ \| $1$ | | |
| $P$ \| $-1$ \| $0$ \| $0$ \| $0$ \| $0$ \| $0$ \| $0$ \| $0$ \| $0$ | A1 | CAO including correct row and column labels |

## Part 8(d):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $p_3 = \dfrac{5}{11}$ | B1 | $p_3$ correctly stated |
| $V \leqslant \dfrac{35}{11},\ V \leqslant \dfrac{35}{11},\ V \leqslant \dfrac{58}{11} \Rightarrow V = \dfrac{35}{11}$ | M1 | Attempt to find $V$ by substituting probabilities into all three equations/inequalities involving $V, p_1, p_2, p_3$ |
| Let Player B play option X with probability $q_1$, option Y with probability $q_2$ and option Z with probability $q_3$ | B1 | Defining probabilities for Player B |
| $5q_2 + 8q_3 = \dfrac{35}{11}$ | dM1 | Dependent on previous M mark; setting up at least three equations in $q_1, q_2, q_3$ using their $V$ |
| $7q_1 + q_2 + 2q_3 = \dfrac{35}{11}$ | | |
| $q_1 + q_2 + q_3 = 1$ | A1ft | Correct three equations (two if Option Z rejected) following through from augmented matrix |
| Player B should play option X with probability $\dfrac{4}{11}$, option Y with probability $\dfrac{7}{11}$ and never play option Z | A1 | Correct options for Player B in context; must state Player B should never play option Z; dependent on previous A mark |

### Alternative for 8(d):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $p_3 = \dfrac{5}{11}$ | B1 | $p_3$ correctly stated |
| Recognises Option Y dominates Option Z; reduces to $3\times2$ or $2\times2$ matrix | M1 | Uses dominance argument; matrix may be transposed and signs changed |
| Define probability for B: play Option X with probability $q$, Option Y with probability $(1-q)$ | B1 | Defining probabilities for Player B (condone use of $p$ instead of $q$) |
| e.g. $-3q + 2(1-q) \Rightarrow 2 - 5q$ and $4q - 2(1-q) \Rightarrow 6q - 2$ | dM1 | Dependent on previous M; setting up at least two equations/expressions in $q$ |
| $2 - 5q = 6q - 2 \Rightarrow q = \dfrac{4}{11}$ | A1ft | Solves to obtain $q$ |
| Player B should play option X with probability $\dfrac{4}{11}$, option Y with probability $\dfrac{7}{11}$ and never play option Z | A1 | Correct options in context; must state never play option Z |

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